Indefinite Integration Questions (389)

Let $f(x)=\displaystyle\int\frac{(2-x^2)\cdot e^x}{(\sqrt{1+x})(1-x)^{3/2}}\,dx$. If $f(0)=0$, then $f\!\left(\dfrac{1}{2}\right)$ is equal to:

\(\int (\tan x + \cot x) dx\) is equal to

Let $f(t)=\displaystyle\int\left(\frac{1-\sin(\log_e t)}{1-\cos(\log_e t)}\right)dt$, $t>1$. If $f(e^{\pi/2})=-e^{\pi/2}$ and $f(e^{\pi/4})=\alpha e^{\pi/4}$, then $\alpha$ equals

If \(\int \frac{1+3\tan x(\tan x + \sec x)}{\tan x} dx = a \log \left|\cos \frac{x}{2} + \sin \frac{x}{2}\right| + C\) where \(0 , then \(a\) is equal to

Given \(f(x) = \int \dfrac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2} dx\). If \(f(0) = 0\), then find the value of \(f(1)\).

If $\int\left[\ln\left(\frac{\cos 2\theta}{1+\sin 2\theta}\right) + \ln\left(\frac{1+\sin 2\theta}{1-\sin 2\theta}\right)^{\cos^2\theta}\right]d\theta$ is equal to $\frac{1}{a}\sin 2\theta\ln\left|\frac{\cos\theta + \sin\theta}{\cos\theta - \sin\theta}\right| + b\ln|\cos 2\theta| + c$ where $a, b \in \mathbb{R} - \{0\}$ & $c$ is integration constant such that $\cos\theta > \sin\theta > 0$ then $(a+b)$ is

\(\int \tan x dx\) is equal to

Let $f$ & $g$ be differentiable function for all $x \in \mathbb{R}$ & have the following properties (i) $f'(x) = f(x) - g(x)$ (ii) $g'(x) = g(x) - f(x)$ (iii) $f(0) = 5$ (iv) $g(0) = 1$ Then the value of $|f(\ln 2) + g(\ln 3)|$ is equal to

If $\int\frac{x^3 + x + 1}{x^4 + x^2 + 1}dx = A_1\ln(x^2 + x + 1) + A_2\tan^{-1}\left(\frac{2x+1}{\sqrt{3}}\right) + A_3\tan^{-1}\left(\frac{2x-1}{\sqrt{3}}\right) + A_4\tan^{-1}\left(\frac{2x^2+1}{\sqrt{3}}\right) + c$ then the value of $(A_1 + A_2 + A_3 + A_4)$ is

Let $\int\frac{\ln\left(x + \sqrt{1+x^2}\right)}{\sqrt{1+x^2}}dx = fog(x) + c$, where $f(x) = \frac{x^2}{2}$ and $g$ are some functions and $c$ is an arbitrary constant. If $\int f(x)g(x)dx = ax^3g(x) + b\left(1+x^2\right)^{3/2} + c\left(1+x^2\right)^{1/2} + d$, then $\left(\frac{1}{a+b+c}\right)$ is equal to

If $\int\frac{(\cos x - \sin x + 1 - x)}{e^x + \sin x + x}dx = \ln(f(x)) + g(x) + c$ where $c$ is the constant of integration & $f(x)$ is positive, then $\frac{f(x) + g(x)}{e^x + \sin x}$ is

If $\int\left(x^{2010} + x^{804} + x^{402}\right)\left(2x^{1008} + 5x^{402} + 10\right)^{10a}dx = \frac{1}{10a}\left(2x^{2010} + 5x^{804} + 10x^{402}\right)^{10a} + c$, where $c$ is constant then $a$ is equal to

If \(\int \dfrac{3\tan\!\left(x - \dfrac{\pi}{4}\right)}{\cos^2 x\,\sqrt{\tan^3 x + \tan^2 x + \tan x}}\, dx = k\tan^{-1}\!\left(\sqrt{\tan x + 1 + \cot x}\right) + C\), then the value of \(k\) is: [where \(C\) is constant of integration.]

If \(\displaystyle\int x^{26}(x-1)^{17}(5x-3)\, dx = \dfrac{x^{27}(x-1)^{18}}{k} + C\), where \(C\) is constant of integration, then the value of \(k\) is:

If \(\int \frac{dx}{\cos^3 x \cdot \sqrt{2\sin 2x}} = (\tan x)^A + C(\tan x)^B + k\), where \(k\) is a constant of integration, then \(A + B + C\) equals

Let \(I_n = \int \tan^n x\, dx\), \((n > 1)\). If \(I_4 + I_6 = a\tan^5 x + bx^5 + C\), where \(C\) is a constant of integration, then the ordered pair \((a, b)\) is equal to

The integral \(\int \frac{dx}{(1+\sqrt{x})\sqrt{x-x^2}}\) is equal to (where \(C\) is a constant of integration)

\(\int (g(x) + 1) \sin x\) dx is equal to

If \(I_n = \int \cot^n x \, dx\) (where \(u = \cot x\)), then the value of \(I_2 + I_3 + I_4 + \ldots + I_9 + I_{10}\) is \(l\)

The integral \(\int \sqrt{1 + 2\cot x(\csc x + \cot x)}\, dx\) is equal to (where \(C\) is a constant of integration):

Evaluate \(\int \frac{\cos x + x \sin x}{x(x + \cos x)} dx\)

\(\int \frac{a+b\cos x}{(b+a\cos x)^2} dx\) is equal to

\(\int \frac{x+1}{x\sqrt{x+1}} dx\) is equal to

\(\int \frac{\sin 2x}{\sin^4 x + \cos^4 x} dx = a\cot^{-1}(b\tan 2x) + c\), then

If \(\int f(x)\,dx = g(x)\), then \(\int f(x^{-1})\,dx\) is equal to

Evaluate $\int \sqrt{\frac{3-x}{3+x}} \cdot \sin^{-1}\left(\frac{1}{\sqrt{6}} \sqrt{3-x}\right) dx$

The integral \(\displaystyle\int \dfrac{dx}{(x+1)^{3/4}(x-2)^{5/4}}\) is equal to

We have \[I = \int\left\{\frac{(\log x - 1)}{1 + (\log x)^2}\right\}^2 dx\] Then \(I\) equals:

Let \[I = \int e^{\sin x}\left(\frac{x\cos^3 x - \sin x}{\cos^2 x}\right)dx\]If \(I = e^{\sin x}\cdot f(x) + C\), then \(f(x) = x\) and find the value of \(\dfrac{f(7)}{2}\).

If \(\int \frac{\sqrt{1-x^2}}{x^4} dx = A(x)(\sqrt{1-x^2})^m + C\), for a suitable chosen integer \(m\) and a function \(A(x)\), where \(C\) is a constant of integration, then \((A(x))^m\) equals:

If \(f(x) = \displaystyle\int \dfrac{(3x^4 - 1)}{(x^4 + x + 1)^2}\, dx\) and \(f(0) = 0\), then \(f(-1)\) is equal to:

\(\int \frac{dx}{\cos x - \sin x}\) is equal to

Evaluate: \int \frac{\cos 2x - 2x \operatorname{cosec}^2 2005}{\sin^2 x \cos x} dx

Evaluate: \int x^5 \sqrt{1+x^3} dx

If \(g(1) = g(2)\), then \(\int_1^2 \frac{[f\{g(x)\}]^{-1} f'\{g(x)\} g'(x)}{f^2(x)} dx\) is equal to

Evaluate: \ 5\int \frac{dx}{\sqrt{1+x} - \sqrt[3]{1+x}}

If $y = f(x) = \frac{3x}{2}$ and $g(x) = f^{-1}(x)$, find $g(1)$ where the curve $y = f^{-1}(x)$ passes through $\left(1, -\frac{2}{3}\right)$

The integral \int \frac{(x+1)^3 + x + x^2 dx}{1 + x} is equal to

Find $\int \ln\left(\frac{x+1}{x-1}\right)^2 dx$

Consider the functions f(x) and g(x), both defined from \mathbb{R} \to \mathbb{R}:f(x) = \frac{x^3}{3} + 1 - x \int_{0}^{x} g(t) dtg(x) = x - \int_{0}^{1} f(t) dtThe minimum value of f(x) is:

Evaluate: $\int \frac{dx}{(2 \sin x + 3 \cos x)^2}$

Evaluate $\int e^{ax} \sin bx dx$ and $\int e^{ax} \cos bx dx$

Evaluate $\int \frac{dx}{3x^2+6x+15}$

Evaluate $\int \sin^{-1} x \, dx$

Evaluate $\int e^{\tan x} (\sec x - \sin x) dx$

Find $\int \frac{dx}{\sqrt{a^2 - x^2}}$

Find $\int \sqrt{3 - 2x - x^2} dx$

Find $\int \sqrt{x^2 + 2x + 5} dx$

Evaluate $\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} \cdot \frac{1}{x} dx$

Find $\int \frac{2x^3 dx}{1+x^2}$